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人気のある微分積分 >

integral of 5/((x-3)(x+2))

解

\int \frac{5}{( x - 3 ) ( x + 2 )} d x

解

ln ∣ x - 3 ∣ - ln ∣ x + 2 ∣ + C

解答ステップ

\int \frac{5}{( x - 3 ) ( x + 2 )} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int \frac{1}{( x - 3 ) ( x + 2 )} d x

以下の部分分数を得る:

\frac{1}{( x - 3 ) ( x + 2 )} : \frac{1}{5 ( x - 3 )} - \frac{1}{5 ( x + 2 )}

= 5 \cdot \int \frac{1}{5 ( x - 3 )} - \frac{1}{5 ( x + 2 )} d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 5 (\int \frac{1}{5 ( x - 3 )} d x - \int \frac{1}{5 ( x + 2 )} d x)

\int \frac{1}{5 ( x - 3 )} d x = \frac{1}{5} ln ∣ x - 3 ∣

\int \frac{1}{5 ( x + 2 )} d x = \frac{1}{5} ln ∣ x + 2 ∣

= 5 (\frac{1}{5} ln ∣ x - 3 ∣ - \frac{1}{5} ln ∣ x + 2 ∣)

簡素化

5 (\frac{1}{5} ln ∣ x - 3 ∣ - \frac{1}{5} ln ∣ x + 2 ∣) : ln ∣ x - 3 ∣ - ln ∣ x + 2 ∣

= ln ∣ x - 3 ∣ - ln ∣ x + 2 ∣

定数を解答に追加する

= ln ∣ x - 3 ∣ - ln ∣ x + 2 ∣ + C

グラフ

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